Below are recommended sources for statistics and data sets related to psychology. This is not an exhaustive list, but can help get you started in your search.
Each policy area page includes relevant organizations, statistical sources, reference sources, and government entities that collect and produce relevant information on the area. These pages are by no means comprehensive, but they will give you an idea of where you might start. Any web search can help you find additional organizations, and a search of usa.gov will help you find relevant government agencies at the federal, state, and local levels.
Policy Areas courtesy of Rebecca Hyde's Social Work Policy & Practice research guide.